k-harmonic Riemannian manifolds

In this work we examine n-dimensional Riemannian manifolds with k-harmonic metrics. Ruse's invariant is shown to be a function of one member of a set of two-point invariants; these are the symmetric polynomials of the eigenvalues of an endomorphism of the tangent space at a fixed point (base point) and of the eigenvalues of the inverse endomorphism. These endomorphisms compare the metric tensor at the base point with the pull-back from a variable point via the exponential mapping. If the k-th symmetric polynomial is a function of the two-point invariant distance function alone, the manifold is k-barmonic at the base point, k-harmonic manifolds are k-harmonic at all base points; thus they form a generalisation of harmonic manifolds. We prove for general Riemannian manifolds: (1) they are harmonic if and only if n-harmonic; (2) all k-harmonic manifolds are Einstein spaces. For simply connected Riemannian symmetric spaces we are able to derive the matrix of the required endomorphism explicitly. We investigate whether these spaces are k-harmonic either for all k or else for no k and prove the former if the rank is one. For symmetric spaces of rank greater than one no firm conclusion is reached.